Theorem fvprif 12781

Description: The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
fvprif	|- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Proof of Theorem fvprif

Step	Hyp	Ref	Expression
1		fvpr0o 12779	. . . . 5 |- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
2	1	3ad2ant1 1019	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
3	2	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
4		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> C = (/) )
5	4	fveq2d 5531	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) )
6	4	iftrued 3553	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> if ( C = (/) , A , B ) = A )
7	3, 5, 6	3eqtr4d 2230	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
8		fvpr1o 12780	. . . . 5 |- ( B e. W -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
9	8	3ad2ant2 1020	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
10	9	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
11		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> C = 1o )
12	11	fveq2d 5531	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) )
13		1n0 6447	. . . . . 6 |- 1o =/= (/)
14	13	neii 2359	. . . . 5 |- -. 1o = (/)
15	11	eqeq1d 2196	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( C = (/) <-> 1o = (/) ) )
16	14, 15	mtbiri 676	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> -. C = (/) )
17	16	iffalsed 3556	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> if ( C = (/) , A , B ) = B )
18	10, 12, 17	3eqtr4d 2230	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
19		elpri 3627	. . . 4 |- ( C e. { (/) , 1o } -> ( C = (/) \/ C = 1o ) )
20		df2o3 6445	. . . 4 |- 2o = { (/) , 1o }
21	19, 20	eleq2s 2282	. . 3 |- ( C e. 2o -> ( C = (/) \/ C = 1o ) )
22	21	3ad2ant3 1021	. 2 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( C = (/) \/ C = 1o ) )
23	7, 18, 22	mpjaodan 799	1 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 979   = wceq 1363   e. wcel 2158   (/)c0 3434   ifcif 3546   {cpr 3605   <.cop 3607   ` cfv 5228   1oc1o 6424   2oc2o 6425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-1o 6431  df-2o 6432

This theorem is referenced by: (None)